Find the constant C such that the function $f(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}$ is a density function of X.

> [!TLDR] > Applies the defining property of a probability density function — the total area under $f(x)$ must equal $1$ — and solves for $C$. > > 1. For $f(x)$ to be a valid probability density function of $X$, it must satisfy $\displaystyle\int_{-\infty}^{\infty}f(x)\,dx=1$. > 2. Here $f(x)=Cx^2$ on $1 $\displaystyle\int_{1}^{4}Cx^2\,dx=1$. > 3. Compute the integral: $\displaystyle\int_1^4Cx^2\,dx=C\left[\dfrac{x^3}{3}\right]_1^4=C\left(\dfrac{4^3}{3}-\dfrac{1^3}{3}\right)=C\left(\dfrac{64-1}{3}\right)=C\cdot\dfrac{63}{3}=21C$. …

Find the constant C such that the function f(x)= Cx² 1<x<4; 0 otherwise is a density function of X.

Question 98 of 105

f(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}

Puducherry TnboardTamil Nadu HSC (DGE) Board 2025Subjective· 2mImportance★★★★★

93% · 98/105 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Applies the defining property of a probability density function — the total area under $f(x)$ must equal $1$ — and solves for $C$ .

For $f(x)$ to be a valid probability density function of $X$ , it must satisfy $\displaystyle\int_{-\infty}^{\infty}f(x)\,dx=1$ .
Here $f(x)=Cx^2$ on $1<x<4$ and $f(x)=0$ elsewhere, so the integral reduces to $\displaystyle\int_{1}^{4}Cx^2\,dx=1$ .
Compute the integral: $\displaystyle\int_1^4Cx^2\,dx=C\left[\dfrac{x^3}{3}\right]_1^4=C\left(\dfrac{4^3}{3}-\dfrac{1^3}{3}\right)=C\left(\dfrac{64-1}{3}\right)=C\cdot\dfrac{63}{3}=21C$ . …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ42 — If in 6 trials, X is a binomial variable which follows the relation 9P…→Next questionQ44 — A lottery with 600 tickets gives one prize of ₹200, four prizes of ₹10…

Q.Find the constant C such that the function $f(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}$ is a density function of X.

Concept understanding — Probability Density Function & Cumulative Distribution

Q.Find the constant C such that the function f(x)={Cx21<x<40otherwisef(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}f(x)={Cx20​1<x<4otherwise​ is a density function of X.

🎓Unlock everything free for 14 days

Q.Find the constant C such that the function $f(x)=\begin{cases}Cx^2 & 1<x<4\\0 & \text{otherwise}\end{cases}$ is a density function of X.

Unlock everything free for 14 days